SG-6 — Moduli stabilization: full dossier

Status (binding, matches the live popup): OPEN (saddle-leaning) — reduction FINISHED, status honestly OPEN. Direction: held. STATUS-UPGRADES:0. Frozen branch dcc66f1b2685 / manifest meta a5b1e6f9d951 READ-ONLY. Serious candidate, NOT validated.

One-line spine. Moduli stabilization is unsolved by every extra-dimensional theory on Earth — and this framework is the rare one that says so out loud, proves the easy half (criticality by symmetry), and points its own loaded gun at itself with a falsifier already half-cocked (the stability-sign decision). This dossier expands that spine into a working-physicist treatment: the rigorous half that is banked, the honest half that is undecided, and a concrete specialist work plan for each open hole.

1. Executive summary + honest status

1.1 Headline

In any higher-dimensional theory the hidden dimensions must sit at fixed sizes and shapes — they must not roll freely — and that resting point must be a true valley (a positive-definite mass matrix), not a mountain pass. Proving this in a fully controlled, global way is a famous open problem that no framework has solved: string, M-theory, F-theory, noncommutative geometry, and lattice constructions all stall on it. It is a shared wall, not a framework-specific weakness.

SG-6's contribution is to split the question into a half it can prove and a half it honestly cannot, and to refuse the one move — anchoring the loop scale at the very stationarity condition that is the electroweak hierarchy — that would let it cheat the hard half.

• The proven half (target-blind, sympy-verified). By a clean group-theory theorem, the selected configuration is automatically a critical point: symmetry pins where the dimensions sit. That theorem is what powers the structural witnesses — integer Wilson-line winding n_H = 1, discrete ℤ₂/ℤ₆ topology, the modular fixed point τ = ω = e^{2πi/3} — assembled into a 12-row witness ledger with a no-tachyon check designed to break if a knob were merely tuned.
• The honest half (undecided). Symmetry is provably silent on whether the critical point is stable. The stability sign is the spectrum, not the symmetry — and the spectrum is the open object.

1.2 The honest grade, in one paragraph

The gate is OPEN (saddle-leaning) because its least-closed residual is open: the moduli-mass Hessian sign is not decided. Two genuine-physics fronts remain. (i) The shape sector was, in earlier 2026-06-25 certificates, banked as a Λ-free −1 saddle; that −1 has since been RETIRED in-corpus as a volume-contaminated, non-critical-point artifact (frozen-branch correction, 2026-06-26), and the properly isolated S₃-irrep shape doublet derives target-blind to mass² {+1,+1} — LEANING STABLE at tree level, NOT refuted and NOT certified. (ii) The breathing/volume singlet rides a shared heat-kernel object (the a₆ Seeley–DeWitt coefficient → the loop scale μ_cell) that is route-INCONSISTENT — two computation routes disagree on value/sign, with one route anchor-falsified — a stronger blocker than "uncomputed." The electroweak-hierarchy carrier, the Hosotani phase θ_H⋆, is read from a one-loop minimum, not derived-small, and its only available anchor (∂_σV = 0) is the hierarchy itself, so using it would be circular. The gate declines that circular save, names the missing experiments, and retracts the artifactual saddle. That is doing science, not marketing.

1.3 What this dossier establishes and does not

Establishes. (a) The witness-type predicate — every downstream-used modulus carries a structural witness of a declared TYPE, so "a potential we minimized at the answer" fails by construction. (b) The orthogonality / criticality theorem — the S₃-fixed point is a critical point of any S₃-invariant functional, target-blind, with zero stability content. (c) A Buckingham-π no-go forbidding a second small mass from {M_Pl, ℏ, dimensionless frozen geometry} without a new σ-carrying length. (d) That the continuum UV divergence of the σ-fluctuation determinant is dissolved by a finite cell-sum (existence of the cost-floor cell transfers cleanly).

Does NOT establish (explicit non-claims, carried verbatim). A positive-definite moduli-mass Hessian (DIAGNOSTIC-ONLY); global stabilization with no flat directions anywhere (explicit NON-claim); a derivation of the electroweak hierarchy (v_EW = 246 GeV is consumed as a measured second ruler, never produced); a fixed value for μ_cell; a Casimir boson-dominance rescue (NOT banked — the chiral index χ = −3 opposes it and the multiplicity table is absent).

Anchors paid. spectrum-E (the SM chiral content + χ = −3, supplied from SG-2/SG-3); the ℏ-footing only for μ_cell (a new floor-value invariant, not the measured ℏ); and v_EW = 246 GeV as a measured-but-irreducible second dimensionful ruler.

2. The community gap

2.1 The precise open problem

A predictive compactification cannot have its hidden shape and size drifting freely: every modulus any later gate reads must be pinned before a number is compared. "Pinned" has a strong and a weak meaning, and the gap is the distance between them:

1. Location (the weak, achievable part). The moduli sit at a definite point of moduli space — a critical point of the effective potential, ∂V = 0.
2. Stability (the hard part). That point is a true minimum: the Hessian ∂²V is positive-definite in every modulus direction (no flat directions, no tachyonic/runaway directions), and this holds in a controlled way — ideally globally over the whole moduli space, not just in a neighborhood.

The community problem is (2) done globally and controllably. Location is comparatively cheap (symmetry often hands it to you); proving the full positive-definite Hessian, with all quantum corrections under control and no runaway at the boundaries of moduli space, is what nobody has delivered.

2.2 State of the art / best bound

This is a genuine textbook SHARED-OPEN wall. The honest cross-framework verdict is TIE / SHARED-OPEN — not a framework win:

• String/flux compactifications stabilize moduli with fluxes and nonperturbative effects (the KKLT and Large-Volume programs), but full control over all moduli, including the de Sitter uplift and the absence of runaways, remains contested — the "swampland" discourse is precisely the admission that controlled, global stabilization is hard and possibly obstructed.
• M-theory / G₂ compactifications face the same moduli-potential control problem.
• F-theory inherits the flux-stabilization difficulties.
• Noncommutative geometry fixes much of the structure spectrally but does not deliver a controlled global moduli potential with a proven positive-definite Hessian.
• Lattice constructions discretize but do not solve the continuum stabilization in closed form.

So the best bound available to anyone is "local/chamber-restricted control under declared assumptions" — which is exactly the honest ceiling SG-6 claims.

2.3 Prior attempts and why each falls short — including this program's own

The most natural attacks, and where each stalls:

• "Write a scalar potential, minimize it, read off the value." This is the hidden-knob trap: a potential whose minimum is placed at the answer is a tuned curve masquerading as a derivation. SG-6's witness-type predicate is explicitly built to make this fail by construction (see §3.1).
• Symmetry alone fixes stability. False, and SG-6 proves it false: a symmetry-fixed point is automatically critical but the symmetry says nothing about the sign of the Hessian. Criticality carries zero stability burden.
• "The directly-computable curvature is negative, so it is a saddle" (this program's own first banked attempt). A 2026-06-25 certificate computed the Λ-free shape-direction scalar-curvature second derivative as d²R/dε² = −1 (raw-ray) / −1/2 (unit-normalized) and banked a SADDLE. The frozen-branch correction (2026-06-26) found this −1 is reproduced exactly but is the raw 3-space Hessian of +R along a volume-contaminated ray (Vol = 1 − ε²) at a non-critical point of R — the gradient grad R(1,1,1) = (−1/2, −1/2, −1/2) ≠ 0. Properly isolated, the shape doublet is an S₃ irrep, so its Hessian is isotropic (∝ I) and a saddle is structurally impossible at the symmetric point. The first attempt fell short by mixing the volume direction into the shape ray. (Lesson logged: a "wall" can be an artifact of the wrong representation — run the faithful-bridge check on the object before grading it.)

The gap that survives all of this is the stability sign, and SG-6's job is to state precisely what is decided, what is undecided, and what would decide it — without anchoring on the answer.

3. The construction — rigorous math

Full common material lives in the published manuscript (Paper I, GUT.html: §5.5, §6.6, Appendix F). This dossier shows the attack-grade derivations and the corrected stability analysis.

3.1 The witness-type predicate (the genuine strong leg)

Gate 6 is not "write a potential, minimize, read off." It is a type-check. Each downstream-used modulus must carry a witness drawn from a declared set of structural mechanisms, and the predicate tests the TYPE of the witness:

 every downstream-used modulus
   --witness-type predicate-->  {Weyl-rigid | modular fixed pt | integer winding | discrete topology | BC pin}
     --TYPE test (NOT minimize-at-answer)-->  PASS (witnessed) / FAIL (tuned-or-floating)
       --12-row ledger + no-tachyon + coverage lint-->  claimed certificate pass (given-E, in-chamber)

Because the predicate tests the type, "a potential we minimized at R₀" fails by construction — it is not a member of the declared witness set. This is the program's honest answer to the hidden-knob attack, and it reproduces by inspection: a reader walks the ledger row by row and confirms each witness is structural, not a number that was tuned.

The declared witnesses and their downstream consumers (the observable-readout map, 10 rows; 5 COMPLETE / 5 BLOCKED_INPUTS):

The frozen geometric data behind these (READ-ONLY, hashes carried for traceability): three Cartan radii 634438ce0776 / 2381d472c62e / 0e8b8dba2cf0; chamber-center (1,1,1) (A1.2); τ = ω 03b30a9c931a; cycle γ + n_H = 1 (A1.12); finite chamber determinant η_BK = 0.009721281516312 84e94518d3f5; RG-transport f531205a9159; comparison scale M_Z a6852c7a6b00; spin-ℂ index / family count χ(K₆,E) = −3 0fd19c9ae0c1; threshold vector (δ₁,δ₂,δ₃) = (+4.8424, −3.1112, −1.7313). θ_H⋆, μ_cell/Δ₀, c_loop, the admissible-rep multiplicity table, and M_R carry no SG-6 hash — read / uncomputed / ABSENT.

3.2 The orthogonality / criticality theorem (banked, target-blind)

Claim. For any S₃-invariant functional V(u₁,u₂,u₃) on the three K₆ Cartan radii, the gradient at the symmetric point (t,t,t) vanishes identically along the invariant direction.

Proof sketch (sympy-verified). The Weyl group S₃ permutes (u₁,u₂,u₃). An S₃-invariant V satisfies V(σ·u) = V(u) for every permutation σ. Differentiate along the totally-symmetric direction at the fixed point (t,t,t): by the chain rule and the invariance, ∂V/∂u_i is the same for all i at (t,t,t), and the antisymmetric combinations that would generate a nonzero shape gradient are forced to cancel by the S₃ symmetry. Hence (t,t,t) is a critical point of V for free — independent of the form of V.

The load-bearing honesty. This is criticality, not stability:

A symmetry-forced critical point is automatically a critical point of any invariant functional, but that fact carries zero stability burden. Location is symmetry; sign is spectrum.

This upgrades the manuscript's "fixed by fiat" wording to "fixed by symmetry as a critical point" — a genuine, free, target-blind structural fact. It does not upgrade to "minimum."

3.3 The symmetry decomposition of the Hessian: 3 = 1 ⊕ 2

The 3-dimensional perturbation space of (u₁,u₂,u₃) around (1,1,1) splits exactly by S₃ into a singlet (breathing/volume, u₁ = u₂ = u₃) and a doublet (shape, traceless). (1,1,1) is a minimum iff both eigenvalues are positive, and they decouple by symmetry:

3.4 The shape-doublet sector — the corrected stability analysis (the load-bearing correction)

This is where the dossier most sharply departs from the older banked result, and the correction is itself the science.

(a) The artifact, reproduced and diagnosed. The flag-manifold scalar curvature along a shape ray u = (1+ε, 1−ε, 1) is, with the normal (equal-scale) metric and R(u) = Σ_i 1/u_i − ½ T(u),

R(ε) = 3/2 − ε²/2.

The ε²-coefficient splits as diagonal convexity +2 (unit-normalized) / +4 (raw-ray) versus the structure-constant triangle term −5/2 / −5, giving a raw second derivative +4 − 5 = −1 and a unit-normalized coefficient +2 − 5/2 = −1/2. (Convention note, carried verbatim from the residual ledger: the two ε²-coefficients +2 and −5/2 sum to −1/2; the raw-ray second derivative is +4 − 5 = −1. A line that writes "+2 − 5/2 = −1" mixes the two conventions and must read either "+2 − 5/2 = −1/2" or "+4 − 5 = −1." The sign is what matters.)

This −1 is reproduced exactly — and it is an artifact. The frozen-branch correction (resolver, 2026-06-26) showed it is the raw 3-space Hessian of +R along a volume-contaminated ray (Vol = 1 − ε²) evaluated at a non-critical point of R: the full gradient is grad R(1,1,1) = (−1/2, −1/2, −1/2) ≠ 0. The ray mixes the volume singlet into the shape direction. The full 3×3 Hessian of R at (1,1,1) has eigenvalues

{ +1 (volume singlet), −1/2, −1/2 (shape doublet) }.

(b) Why a shape saddle is structurally impossible. Once the shape doublet is properly isolated (orthonormal trace-free log basis, fixed volume, shape-critical), it is an S₃ irrep, so its Hessian is isotropic (∝ I). Two equal eigenvalues cannot have opposite signs — therefore a saddle is structurally impossible at the symmetric point. The doublet is nonetheless a genuine physical modulus: R changes along it, the residual symmetry is the discrete Weyl group S₃ (no continuous orbit to quotient away), and the shape-sector field-space metric G_shape = I is positive-definite. So the −1 is not a redundant-direction artifact either; it is specifically a wrong-ray/non-critical-point artifact.

(c) The decisive datum: the Einstein-frame potential sign. The physical mass² in the shape sector is M = G⁻¹ · Hess(V) on the doublet, and the verdict is fixed entirely by the sign of the Einstein-frame shape-sector potential:

V = +R  →  mass² {−1, −1}   (shape MAXIMUM / unstable)
V = −R  →  mass² {+1, +1}   (shape MINIMUM / stable)

The frozen-branch derivation (2026-06-26, sympy-verified, with specialist cross-check) finds the Einstein-frame shape sign target-blind as V_phys ~ −R_K6, hence physical mass² {+1, +1} → the symmetric flag u = (1,1,1) is a STABLE shape MINIMUM at tree level. A capability-to-fail control on S²×S² fires the opposite sign (+16), confirming the test can fail; the Weyl step is sign-neutral.

(d) The honest caveats — why this is LEANING STABLE, not certified. Three things keep this from being a certificate: 1. Independence is partial. The two routes share the additive-curvature-split + G_Newton > 0 core, so this is one anchored derivation + a cross-check, not a strict two-independent-route value (the closure-path step-7 gate is not strictly met). 2. Full-potential-completeness is OPEN. Is ±R the whole shape-sector object, or are there additional contributions (loop/Casimir, higher-curvature)? That completeness theorem is unproven — it is the shape sector's own blocker, distinct from the heat-kernel wall. 3. Checklist-completeness is OPEN. The shape-sector checklist-completeness theorem remains unproven.

So the corrected shape-sector verdict is NOT a saddle, NOT a certified minimum — LEANING STABLE, blocked on two completeness theorems plus the loop/Casimir sign. Governance caught a worker swarm over-promoting this to CERTIFIED_LOCAL_SHAPE_MINIMUM across the unpinned sign and refused certification — the anti-promotion discipline working internally.

3.5 The breathing/volume singlet — the heat-kernel wall (the genuine open core)

The breathing-singlet eigenvalue rides V''(σ) at the e^{−6σ} term, i.e.

c_loop = tr[a₆]   (the Seeley–DeWitt heat-kernel coefficient of the σ-fluctuation determinant on the 6-manifold K₆),

whose log-scheme residue is a single scale factor μ_cell. This is the deep object shared with Gap-01 (the a₆ / UV input) and SG-7 (the threshold normalization). Two findings sharpen its status well past "uncomputed":

• The continuum divergence is dissolved (banked). Replacing the continuum mode integral by a finite cell-sum under the uniform operational cell law N ≤ B/Δ₀ genuinely kills the UV (a → 0) divergence — the continuum-as-source-of-divergence assumption is false, and dropping it is banked (the granularity root). But finiteness does not imply uniqueness: a finite supertrace of a log-running 6-D determinant still carries one log-scheme degree of freedom, μ_cell.
• The object is route-INCONSISTENT (2026-06-27 engine update). A two-route cross-check of the comparable scale-free K₆ vector/ghost a₆/a₀ object disagreed: Route A (Gilkey contraction) vs Route B (Gilkey-free spectral) differ by |31/48| ≈ 0.65 — about 6 orders outside the 1e-6 tolerance — so the binding two-route value rule FAILED. The fault-localizer is decisive: at the K₆-bundle anchor, Route B reproduces the canonical vector a₆/a₀ = −16/315 Gilkey-free (diff 2.7e-9, CORRECT), while Route A returns −43/504 (MATCH = False) — Route A's K₆-bundle a₆ sector is anchor-falsified. Route B dissolved the named su(3) root-shell blocker for the ghost (ghost LC a₆/a₀ = 149/1008), but the graviton Sym²(T) sector is still missing (4-band mixing). So the obstruction is upstream of the μ_cell scheme residue: even before the (forbidden) ∂_σV = 0 anchor question, the two routes do not agree on the a₆ value/sign.

A retraction the program owns plainly. An earlier a₆-dependent dimensionful value, −2.817995812e94 GeV⁶, was RETRACTED as 31/147-contaminated; the corrected consistency coefficient −2.995681680e94 GeV⁶ is scheme-anchored only (never gap-closing). The curvature input was also corrected: the Bianchi-exact ratio is |Riem|²/Scal² = 23/75 (residual ~3e-16); the engine's earlier 31/147 was Bianchi-violating. The "~31% route disagreement" sometimes quoted is the benign face of this; the load-bearing face is the value/sign route-inconsistency above. Naming a withdrawn value is part of the honest ledger, not a footnote.

3.6 The electroweak-hierarchy carrier θ_H⋆ — read, not derived

The Hosotani phase θ_H⋆ ≈ 2.46×10⁻¹⁴ carries ~85% of the electroweak hierarchy via

v_EW = θ_H⋆ / (2π R_γ).

It is extracted from the one-loop V_Hos minimum, not computed-small from chamber data. The frozen η_BK = 0.009721 gives only √η_BK / 2π ~ 10⁻² — about 12 orders short. Two derivation routes, both killed by the firewall:

The legitimate banked sub-result: the Higgs-mass quadratic sensitivity is finite/protected — but only to ~10¹⁴ GeV, ~12 orders short of the full hierarchy, and recorded as such, not promoted.

3.7 The Buckingham-π second-mass no-go (banked)

At the dimensional-counting level: from {M_Pl, ℏ, dimensionless frozen geometry} you cannot construct a parametrically small second mass without a length that carries the relevant scale. This is the structural reason the hierarchy must be read, not deduced, in this geometry — and it is what makes v_EW = 246 GeV a measured-but-irreducible second ruler (same category as Λ and η_B), consumed as an anchor and never produced.

4. The insights we used

These are the moves that made the progress believable and reproducible — shared so a reader can check and extend them.

1. Witness-TYPE over witness-value. The decisive design choice is to test the kind of pin, not its number. A type-check makes "minimize-at-answer" inadmissible by construction, which is a far stronger anti-tuning guarantee than any sensitivity bound. This is the y = R² curve-not-a-number discipline applied to every dial the certificates touch.

2. Location is symmetry; sign is spectrum. Separating criticality (free, from S₃) from stability (the Hessian sign, the real work) is the single clarifying insight. It immediately tells you which witnesses are cheap (the fixed points) and which carry the whole burden (the eigenvalue signs), and it forbids the common error of citing a symmetry-fixed point as evidence of a minimum.

3. Decompose the Hessian by the residual symmetry: 3 = 1 ⊕ 2. The S₃ split decouples the volume singlet from the shape doublet exactly, so each can be attacked with the right tool: the doublet by Λ-free flag-manifold curvature, the singlet by the heat-kernel determinant. This is what made the shape sector computable in isolation — and, crucially, what exposed the volume-contamination artifact when the two were mixed.

4. Isotropy of an irrep Hessian is a structural theorem, not a computation. Because the shape doublet is a single S₃ irrep, its Hessian must be proportional to the identity — so a saddle is structurally impossible, independent of any numerical curvature value. Recognizing this is what turned a banked "−1 saddle" into a diagnosed artifact: the −1 could not be the shape doublet's eigenvalue, so it had to be measuring something else (the volume-contaminated ray at a non-critical point).

5. The sign is the whole verdict; the magnitude is a distraction. In the shape sector, V = ±R decides MIN vs MAX entirely by sign. The persistent magnitude route-disagreement is non-load-bearing for the stability question — a focusing insight that isolates the one datum worth deriving (the Einstein-frame potential sign).

6. Collapse the hard fronts onto one object, then firewall it. R1 (the hierarchy via θ_H⋆), R3/R6 (the doublet Casimir magnitude), and R5 (the breathing singlet c_loop) all reduce to one shared object: μ_cell, the spectral value of the uniform operational cell Δ₀. The κ³/π falsification test then asks the single decisive question — is there a v-independent readout of μ_cell? — and answers no: its only anchor (∂_σV = 0) IS the hierarchy. Reducing three problems to one and then proving that one cannot be closed without circularity is the honest endpoint, not a failure.

7. Faithful-bridge check before grading. The lesson from the retired saddle: a "wall" can be an artifact of the wrong representation. Before grading an object OPEN or banking a verdict, check that the object you computed is the object you meant (right ray, critical point, correct heat-kernel representation for a global ℤ₂ orbifold). This caught both the volume-contaminated −1 and a wrong boundary-value-problem a₆ representation upstream.

5. Evidence & reproducibility

5.1 What reproduces, and how to re-run it

• The witness-type discipline reproduces by inspection. Walk the 12-row ledger (Appendix F.3); confirm each witness is structural (admissibility / symmetry fixed-point / integer winding / discrete topology / BC pin) and not "a number we minimized to." No computation needed.
• The discrete/topological witnesses reproduce by hand. n_H = (1/2π)∮_γ F = 1, the ℤ₂ orbifold y → −y, the ℤ₆ identification, the cycle γ — all topological invariants, nothing to drift.
• The S₃-criticality theorem reproduces symbolically. Build any S₃-invariant V(u₁,u₂,u₃); verify ∂V along the invariant direction vanishes at (t,t,t) by the chain rule. sympy-verified, target-blind.
• The corrected shape sector reproduces symbolically. (i) R(ε) = 3/2 − ε²/2 along u = (1+ε,1−ε,1). (ii) grad R(1,1,1) = (−1/2,−1/2,−1/2) ≠ 0 — the ray is non-critical, the −1 is an artifact. (iii) Full Hess R(1,1,1) eigenvalues {+1, −1/2, −1/2}. (iv) On the isolated trace-free doublet, V = −R → mass² {+1,+1} (LEANING STABLE); the S²×S² control fires +16 (opposite), proving the test can fail. Do this in a scratch directory; the frozen branch is READ-ONLY.
• The generic modular fixed point reproduces symbolically. The order-three elliptic element M = [[0,−1],[1,−1]] (trace −1, det 1) of PSL(2,ℤ) fixes exactly one point of the upper half-plane, ω = e^{2πi/3}.
• The Buckingham-π no-go reproduces by dimensional counting. No second mass from {M_Pl, ℏ, dimensionless geometry} without a σ-carrying length.

5.2 What does NOT reproduce (honest gaps)

• The full moduli-mass Hessian sign — the shape sector is LEANING STABLE (not certified, two completeness theorems open); the breathing singlet is route-inconsistent at the a₆ level.
• θ_H⋆ as a derivation — read from a minimum; no SG-6 hash.
• μ_cell's value — a new unfixed invariant; no v-independent readout (the only anchor is forbidden).
• The Casimir net sign — needs the ABSENT admissible-rep multiplicity table; boson-dominance NOT banked (χ = −3 opposes).
• Phenomenological sufficiency at the numerical level — inherits the SG-7 threshold harness, which is itself AUDIT (reproduce_all.py confirmed absent corpus-wide; the δ-vector magnitudes are injected reals).

5.3 The no-target-fitting audit (8 checks, all PASS)

The discipline guard is itself a certificate. All eight checks PASS:

1. μ_cell never anchored at ∂_σV = 0 (it stays AXIOM-OPEN).
2. No Casimir rescue banked without the multiplicity table (on-disk: degeneracies_known=false, sign_stable=false, analytic_continuation_present=false; backsolve guard all-false).
3. Existence/value split kept explicit (only the existence of Δ₀ transfers; the value μ_cell stays open).
4. The single ATOMIC leg v_EW = 246 GeV is a measured invariant, not an axiom-reproduced number.
5. Non-merge wall held — v_EW never folded into μ_cell.
6. Criticality not banked as stability (the S₃ and τ=ω fixed points carry zero stability burden).
7. The frozen branch is untouched (READ-ONLY; verification in scratch only).
8. δ-vector magnitudes not laundered into a derivation (R9 stays BLOCKED_INPUTS).

The audit schema independently SCHEMA_CONVERGED with STATUS-UPGRADES:0, with truth-promotion firewalled off — audit-done does NOT imply theory-confirmed (the BS-8 discipline: a clean audit / SCHEMA_CONVERGED is not evidence the theory is true).

5.4 Frozen-object hash index (load-bearing, READ-ONLY)

Branch dcc66f1b2685 · manifest meta a5b1e6f9d951 · three Cartan radii 634438ce0776 / 2381d472c62e / 0e8b8dba2cf0 · chamber-center (1,1,1) (A1.2) · τ = ω 03b30a9c931a · η_BK = 0.009721281516312 84e94518d3f5 · RG transport f531205a9159 · M_Z a6852c7a6b00 · spin-ℂ index χ = −3 0fd19c9ae0c1 · threshold vector (+4.8424, −3.1112, −1.7313). θ_H⋆, μ_cell/Δ₀, c_loop, the multiplicity table, and M_R: no SG-6 hash — read / uncomputed / ABSENT.

6. Open gaps + closure path (the specialist work plan)

Each hole is a work-package. The two genuine-physics fronts (the hierarchy and the Hessian sign) collapse onto one shared object — μ_cell — and the firewall has already shown it has no v-independent readout, so every μ_cell-touching path carries the explicit κ³/π falsification test: the ∂_σV = 0 anchor is forbidden (it IS the hierarchy). Realistic campaign outcome: ~3 Reduced-to-Axiom, ~2 DISCLOSED-CONSISTENT, the Hessian-sign decision pending owner artifacts. No DERIVED-CLOSED is promised; STATUS-UPGRADES:0. The single most valuable likely outcome is a NEGATIVE: a confirmed instability firing the no-minimum falsifier.

6.1 Hole R3/R6 — the saddle-vs-minimum decision (the Hessian sign)

(a) Precise statement. Decide the physical-mass² sign of the full moduli-mass Hessian at (1,1,1). The shape doublet is LEANING STABLE pending two completeness theorems; the decisive remaining datum is the graded (boson − fermion) Casimir net sign, which would either confirm stability or flip the verdict. The exact missing object: the admissible-rep multiplicity table for the K₆ spectrum and a regularization-stable s = −1 zeta continuation (equivalently the d = 6 scalar Seeley–DeWitt a₄ coefficient).

(b) Why it's hard / prior-attempt lessons. The first banked attempt mistook a volume-contaminated −1 for a shape saddle — do not repeat that: compute on the isolated trace-free shape doublet at a critical point, in an orthonormal log basis. The per-sector Casimir signs are LOCKED (bosons stabilize +, fermions destabilize −, scheme-independent by convexity g''(0) = 2(w₁² + w₂²) > 0 and f'(m²) > 0); the net sign is sgn(boson-surplus − fermion-surplus) over admissible (p,q). Trap to avoid: banking "boson-dominated rescue" from the ~3:1 heat-kernel ratio — that is the κ³/π-forbidden plausible-but-unverified sign, and χ = −3 opposes it. On-disk the only multiplicity artifact is a placeholder marked MISSING; the backsolve guard reads all-false. Do not bank a rescue without the table.

(c) Exactly what closes it. Mount the multiplicity table; form the graded doublet-weighted supertrace Σ (−1)^F d·n·(w₁² + w₂²) with the regularization-stable s = −1 continuation; decide the net sign and the magnitude μ_cell·q (≳ 0.5 needed to overcome any residual curvature). Success criterion: net > 0 and μ_cell·q ≳ 0.5 → VERIFIED-RESCUE / minimum. A REFUTING result is equally valid: net ≤ 0 → SADDLE-CONFIRMED, the F.9.7(2) falsifier fires, and the moduli-mass-generation row downgrades. Note: a closed-form multiplicity m₀(p,q) = min(p,q)+1 if (p−q) ≡ 0 mod 3 else 0 verifies against textbook values (adjoint → 2, (2,2) → 3, (3,3) → 4, complex reps → 0) — one derivation step done, but R6 stays blocked on the second artifact (the regularized zeta) and the v-free μ_cell magnitude.

(d) Machinery & inputs. Lie-theory of SU(3)/T² representations; zeta-function regularization / Seeley–DeWitt a₄ on a 6-manifold; the KK eigenvalue spectrum. Corpus starting points: TEST_SHAPE_DOUBLET_STABILITY_2026-06-23.md, TEST_CASIMIR_RESCUE_SHAPE_DOUBLET_2026-06-23.md, the F1 resolution cert SHARED_FINDINGS/F1_shape_doublet_saddle/, and the R6 decision cert SG6_completion_endpoint/05_R6_CASIMIR_DECISION.md.

(e) Leverage. Decides SG-6 R3/R8 directly; the same Einstein-frame sign datum decides UQF-10 S-1/S-2 and the F1 cross-gate finding. Mounting the multiplicity table is the single highest physics value-per-effort move — it converts a diagnostic into a falsifiable verdict with no new tuning.

6.2 Hole R5 — the breathing-singlet c_loop / a₆ heat-kernel wall (route-inconsistent)

(a) Precise statement. Compute c_loop = tr[a₆] of the σ-fluctuation determinant on the frozen K₆ branch, target-blind, fixing the singlet Hessian eigenvalue sign; equivalently, reconcile the two routes for the scale-free a₆ object and then fix μ_cell from an independent footing-of-ℏ input.

(b) Why it's hard / prior-attempt lessons. This is the deep 13D FRG-2 β-vector wall, shared with Gap-01 and SG-7. The object is route-INCONSISTENT, not merely uncomputed: Route A vs Route B disagree on the comparable K₆ vector/ghost a₆/a₀ by |31/48| ≈ 0.65 (~6 orders), and Route A is anchor-falsified (returns −43/504 vs the canonical −16/315). Traps to avoid: (i) do not quote the "~31% route disagreement" as if it were a benign finite-coefficient spread — the load-bearing fault is a value/sign inconsistency, and a 31/147-contaminated dimensionful value (−2.818e94 GeV⁶) was retracted; (ii) do not use the wrong heat-kernel representation — a global ℤ₂ reflection orbifold needs the Donnelly equivariant (Lefschetz) defect, not a manifold-with-boundary mixed-BVP a₆ tower; (iii) never anchor μ_cell at ∂_σV = 0 (circular). Use the Bianchi-exact curvature input |Riem|²/Scal² = 23/75 (not 31/147).

(c) Exactly what closes it. (i) Fix Route A's K₆-bundle a₆ sector against a K₆-bundle spectral peel; (ii) deliver Route B's graviton Sym²(T) LC a₆ via moment-polynomial / spectral-zeta resummation (the ghost sector is already dissolved: ghost LC a₆/a₀ = 149/1008); (iii) re-run the binding two-route value rule. Success: the two routes agree within 1e-6 → c_loop sign decided → singlet eigenvalue fixed. REFUTING result: a definite c_loop of the wrong sign → the breathing singlet is unstable → (1,1,1) is not a minimum (a valid close). Short of the full computation, the honest endpoint is Reduced-to-Axiom at AXIOM-UNIFORM-CELL-VALUE (Δ₀ existence is the granularity root; the spectral value is a floor-value residue on the footing of ℏ).

(d) Machinery & inputs. Functional renormalization group; Seeley–DeWitt/Gilkey a₆ on a 6-manifold; su(3) Gelfand–Tsetlin matrix elements for the Levi-Civita-vs-canonical correction; spectral-zeta resummation. Corpus: TEST_CLOOP_CELLSUM_ATTACK_2026-06-23.md, the B1 heat-kernel build SHARED_BLOCKER_BUILDS/B1_heatkernel/, and the route-inconsistency note in SG6_COMPLETION_HANDOFF/01_DOSSIER.md (cross-gate propagation note).

(e) Leverage. It is ONE object load-bearing for three gates: Gap-01 (a₆ value/sign), SG-6 (μ_cell → the R3 breathing-singlet eigenvalue), and SG-7 (threshold normalization). Reconcile the two routes once — fix the su(3) GT off-diagonal / Lichnerowicz-hopping matrix elements and the ghost discrepancy target-blind — and the fix propagates to all three. (This touches only the singlet leg; the shape doublet is a separate Λ-free fact.)

6.3 Hole R7 — τ = ω realization-specific modular uniqueness

(a) Precise statement. Prove τ = ω = e^{2πi/3} is the unique order-three fixed locus of the realized modular generator on the F⁺ Cartan-torus group within the Weyl-rigid chamber — independent of any one-loop potential.

(b) Why it's hard / prior-attempt lessons. The generic PSL(2,ℤ) elliptic-uniqueness fact is proven (one fixed point per order-three element), but it does not by itself establish uniqueness for the specific realized generator inside the chamber. F.2's "perturbations off ω produce a non-zero potential" leans on a potential that is not computed — the leg should rest on symmetry uniqueness, not an uncomputed curve. Trap to avoid: do not inject any flavor or hierarchy number; this must be pure group theory.

(c) Exactly what closes it. Enumerate the fixed points of the realized modular group on the upper half-plane; confirm ω is the order-three fixed point and characterize uniqueness within the chamber. Success: uniqueness proven → R7 upgrades from Reduced-to-Axiom to DERIVED, converting the τ-witness from "potential we did not compute" to "symmetry-FORCED." REFUTING result: a second fixed locus exists → the witness is non-unique (sharper-OPEN).

(d) Machinery & inputs. Finite group theory of the F⁺ Cartan-torus modular action; PSL(2,ℤ) elliptic-element fixed-point theory. Corpus: 06_R7_MODULAR_FIXED_POINT.md.

(e) Leverage. The single cleanly-upgradable leg in SG-6 (contained, bounded, no target). Also hardens SG-8, which inherits the τ = ω soft spot (operator diagonality + CP phase depend on it).

6.4 Hole R1 — θ_H⋆ derived target-blind (the EW-hierarchy relocation half)

(a) Precise statement. Derive θ_H⋆ target-blind from frozen chamber data {γ, n_H = 1, η_BK, R_γ} plus a UV reference frozen before and independently of v_obs, and check whether it returns ≈ 2.46×10⁻¹⁴ with no v in the inputs.

(b) Why it's hard / prior-attempt lessons. Path A (read off V_Hos) is circular at ∂_σV = 0; Path B (transmutation exponent) is the wrong mechanism (θ_H⋆ is a periodic-potential minimum location, not a running coupling). The Buckingham-π no-go blocks a second mass from {M_Pl, ℏ, shape}. Trap to avoid: do not pin μ_cell from the EW hierarchy and call it a prediction — that has relocated, not removed, the input.

(c) Exactly what closes it. Write V_Hos(θ_H) from the frozen cycle/winding/η_BK data; locate its minimum without consulting v_obs; check whether it lands at ~10⁻¹⁴ from chamber data alone or only after a scale is pinned by v. Success: target-blind θ_H⋆ ≈ 10⁻¹⁴ → the hierarchy's dominant factor delivered (unlikely — Buckingham-π blocks it). Realistic endpoint: Reduced-to-Axiom at AXIOM-VEW-SECOND-ANCHOR — v_EW conceded as a second irreducible dimensionful ruler with a principled (dimensional + mechanistic) reason. REFUTING result: a chamber-only θ_H⋆ that misses 10⁻¹⁴ → a genuine prediction-vs-data failure of the lightness-of-v leg.

(d) Machinery & inputs. Hosotani/Wilson-line effective-potential mechanics; Buckingham-π dimensional analysis. Corpus: SCALE_HIERARCHY_FINAL_VERDICT_2026-06-23.md, CLOSURE_CAMPAIGN_RESULT_2026-06-24.md, readout-map row 1.

(e) Leverage. Carries ~85% of the EW hierarchy. Until derived, v_EW stays the ATOMIC anchor and θ_H⋆-as-derived stays OPEN/RELOCATION; the principled second-anchor statement is the honest ceiling, not a higher claim.

6.5 Hole R9 — phenomenological sufficiency (downstream AUDIT, owned by SG-7)

(a) Precise statement. Confirm that residual moduli leave every Gate 1–10 output within its published tolerance, machine-real, by propagating an SG-6 radius/threshold perturbation through the downstream pipeline.

(b) Why it's hard / prior-attempt lessons. It relies on the Gate-7 threshold pipeline, which is itself AUDIT: the δ-vector (+4.8424, −3.1112, −1.7313) has geometric signs but fitted-to-target magnitudes (injected reals), and reproduce_all.py + the ledger CSVs are confirmed absent corpus-wide. Trap to avoid: do not treat the injected δ as derived, and do not reverse-engineer a normalization to hit a known δ (that relocates the fit).

(c) Exactly what closes it. Mount appendix_F_heat_kernel_ledger.csv and appendix_F_threshold_outputs.csv; run reproduce_all.py against the frozen radii + spectrum target-blind; confirm the threshold vector + unification residual regenerate; confirm a small δ on the SG-6 radii moves the Gate-7 output within the F.3 δ-response column. Success: VERIFIED (sufficiency machine-real). REFUTING result: a perturbation breaches tolerance and is not bounded by F.3 → sufficiency fails and the offending downstream gate downgrades.

(d) Machinery & inputs. The SG-7 reproduction harness (execution + verification, no new physics). Corpus: the readout-map δ row; SG-7's own hole queue.

(e) Leverage. Closing SG-7's δ-harness AUDIT closes this too — shared, not independent.

6.6 The disclosed residuals (kept honest, no flip available)

• R2 — global stabilization is an explicit NON-claim (DISCLOSED-CONSISTENT). Off-chamber configurations are rejected by admissibility, not dynamically stabilized. This is the SHARED-OPEN textbook problem; no SG-6 axiom flips it. Keep the F.10.2 vocabulary rule enforced (ban "full stabilization" / "no flat directions" without the chamber qualifier) and never let a downstream V(σ) cross-reference upgrade this gate's certificate.
• R4 — given-E / within-chamber (DISCLOSED-CONSISTENT / relocation). The witnesses fix the moduli inside the already-selected chamber; selection ≠ derivation, given-E ≠ derivation of E. This is the load-bearing conditionality of the whole gate; nothing at SG-6 closes it (it relocates to SG-1's SHAPE-forcedness and SG-3's given-E). The one contained strengthening is R7.
• R8 — no-tachyon (DISCLOSED-CONSISTENT, conditional on R3/R5). The verdict is the AND of the two physical-eigenvalue rows; it cannot be made target-blind without the R6 multiplicity table and the R5 c_loop (both absent). A confirmed shape instability is the field-space analog of a tachyon and would put pressure here.

7. Honest ceiling & scope

Dissolved ≠ solved. The cell-sum dissolves the continuum UV divergence of the σ-fluctuation determinant — a genuine, value-free win — but dissolution is not closure: a finite supertrace still carries one log-scheme factor μ_cell, and the value question survives.

Selection ≠ derivation. The witnesses certify that the selected geometry is internally controlled; they do not prove the witnesses determine the geometry. The chamber, the bundle E, and χ = −3 are inputs from upstream.

Given-E ≠ derivation-of-E. SG-6 operates on the supplied SM chiral content; it never derives it.

Criticality ≠ stability. The S₃-fixed point and τ = ω are critical points forced by symmetry, carrying zero stability burden. The stability content is the Hessian sign — and that is the open object.

Reduced-to-Axiom ≠ proven. The named axiom floors (AXIOM-VEW-SECOND-ANCHOR, AXIOM-UNIFORM-CELL-VALUE, AXIOM-MODULAR-FIXED-POINT) name where the reduction lands; they are bookkeeping of the honest bottom, not proofs.

The dissolved unicorn (a shared ceiling, not a framework gap). A fully controlled, GLOBAL moduli stabilization with a derived positive-definite mass matrix valid over the entire moduli space is an explicit NON-claim and a shared-open textbook problem that no higher-dimensional framework (string, M-, F-theory, noncommutative geometry, lattice) has solved. The chamber-restricted certificate is the honest ceiling; the global version is unprovable-in-practice for everyone — a limit on all current higher-dimensional physics, not a framework-specific weakness.

What is explicitly NOT claimed. A positive-definite Hessian (DIAGNOSTIC-ONLY); a certified shape minimum (LEANING STABLE only, two completeness theorems open); a derivation of the electroweak hierarchy or of θ_H⋆ (read from a minimum); a fixed μ_cell value; a Casimir boson-dominance rescue (χ = −3 opposes; multiplicity table absent); global stabilization.

The anchors paid. spectrum-E (SM chiral content + χ = −3); the ℏ-footing only for μ_cell (a new floor-value invariant, not the measured ℏ); v_EW = 246 GeV as a measured-but-irreducible second ruler. No clean termination on M_Pl / α_i(M_Z) / y_t / |V_us|.

The honest one-sentence endpoint. SG-6 is complete as an audit but OPEN as physics: the structural-witness grammar is a genuine target-blind method/group-theory win carrying zero stability burden; the earlier −1 shape saddle is retired as a volume-contaminated artifact and the isolated shape doublet leans STABLE ({+1,+1}, not certified); the genuinely undecided core is the route-inconsistent a₆ breathing-singlet sign and the Casimir net sign (blocked on an absent multiplicity table, with χ = −3 opposing the boson rescue); μ_cell is never anchored at the forbidden ∂_σV = 0; and the single most valuable likely outcome remains a NEGATIVE — an instability firing the no-minimum falsifier. STATUS-UPGRADES:0; frozen branch dcc66f1b2685 / a5b1e6f9d951 READ-ONLY; serious candidate, NOT validated; nothing applied, nothing deployed.

Source map (corpus inputs actually read for this dossier)

• …/TOE/PER_GATE_DOSSIERS/SG6_COMPLETION_HANDOFF/01_DOSSIER.md — the per-gate closure-attack dossier + cross-gate propagation note (route-inconsistent a₆).
• …/TOE/PER_GATE_DOSSIERS/SG6_COMPLETION_HANDOFF/certificates/SG6_completion_endpoint/ — 02_OBSERVABLE_READOUT_MAP.csv, 03_RESIDUAL_LEDGER_R1_R9.md, 04_R3_SHAPE_DOUBLET_SADDLE.md, 05_R6_CASIMIR_DECISION.md, 06_R7_MODULAR_FIXED_POINT.md, 08_NO_TARGET_LOADING_AUDIT.md.
• …/TOE/PER_GATE_DOSSIERS/SG6_COMPLETION_HANDOFF/08_SG6_ENDPOINT_PASS_2026-06-25.md.
• …/TOE/PER_GATE_DOSSIERS/SG6_COMPLETION_HANDOFF/ axioms-and-blind-assumptions ledger — F1 RESOLVED + SIGN-RESOLVED (shape {+1,+1}); B1-RECLASSIFICATION + retracted −2.818e94 GeV⁶; B1-R6 ENGINE UPDATE (route inconsistency |31/48|, Route A anchor-falsified −43/504 vs −16/315); SG-6 per-gate inventory.
• …/TOE/PER_GATE_DOSSIERS/SHARED_FINDINGS/F1_shape_doublet_saddle/03_RESOLUTION.md — the artifact diagnosis + Einstein-frame sign decision.
• …/TOE/GATE_REGRADE_BESTCASE_AND_HOLES_2026-06-29.md — the SG-6 regrade (binding status string + review note retiring the saddle).
• …/TOE/SPECIALIST_HOLE_QUEUE_2026-06-29.md — the 5-hole SG-6 work queue.
• …/TOE/30pagedoc/handoffs/SG6.md — the build handoff (honest status, banked, bright-lines, holes).

Dossier built 2026-06-29. Our geometry (13D K₆ branch) only. Common derivation material referenced to the published manuscript (Paper I, GUT.html §5.5 / §6.6 / Appendix F), not duplicated. STATUS-UPGRADES:0.